What is the significance of a $\chi^2$ analysis? What is the merit of $\chi^2$ analysis in the laboratory experiments? Why is so much emphasis put on it? 
 A: To test a model of the form $y=f_a(x)$ where $a$ parameterises a function in a family, finding best-fit $a$ from experimental data, we don't try to solve simultaneous equations $y_i = f_a (x_i)$ by assuming every data point is exact. Measurement error, random noise etc. means that's physically unrealistic, and you'd almost certainly find the equations are inconsistent anyway. Instead the data point $(x_i, y_i)$ satisfies $y_i = f_a (x_i)+\epsilon_i$.
In theory, one can find some collective measure of how large the error terms $\epsilon_i$ are, to assess (i) how to minimise it with the choice of $a$ and (ii) whether the result is good enough to believe the model. There are theoretical reasons to expect the $\epsilon_i$ to have mean-$0$ Normal distributions, and if we think we know the standard deviation is $\sigma$ then for $n$ data points we have $\sigma^{-2}\sum_i\epsilon_i^2\sim\chi_n^2$. The great thing about knowing this quadratic function's distribution is we can now express the empirical result's "weirdness" as a $p$-value.
Why do we use this function? Because it's easy. You can't use $\sum_i\epsilon_i$, because negative errors need to penalised rather than rewarded. Nor do we use $\sum_i|\epsilon_i|$, because working out the distribution of that is a lot harder. What's more, for many popular problems it's easy to compute which $a$ minimises $\sum_i\epsilon_i^2$. For example, any $f_a$ of the form $\sum_j a_j f_j(x)$ reduces this problem to matrix inversion.
